Use implicit differentiation to find , assuming in the case that y is a function of x.
Using the chain rule on the left-hand side, and the fact that the derivative of x with respect to x is 1 on the right-hand side, we find
To find the derivative of xy we need to use the product rule.
Multiply both sides by xy to remove that pesky denominator, then move all the terms on one side and solve:
is also correct, since this is the result of multiplying our first answer by .
Now move all the terms on one side, factor, and solve:
Multiply both sides by x2 to remove the denominator:
Then move all the terms with to one side, factor, and solve:
Make it rain.
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